Spectral Theorem approach to the Characteristic Function of Quantum Observables

Using the spectral theorem we compute the Quantum Fourier Transform (or Vacuum Characteristic Function) $\langle Φ, e^{itH}Φ\rangle$ of an observable $H$ defined as a self-adjoint sum of the generators of a finite-dimensional Lie algebra, where $Φ$ is a unit vector in a Hilbert space $\mathcal{H}$. We show how Stone's formula for computing the spectral resolution of a Hilbert space self-adjoint operator, can serve as an alternative to the traditional reliance on splitting (or disentanglement) formulas for the operator exponential.

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